Manipulatives and number sentences in computer aided arithmetic word problem solving

In this study we investigated the relative contribution of two main components often used in the instruction of arithmetic word problem solving to first-grade children and children with learning problems, external representation with manipulatives and formal mathematical representation with number sentences. Four computer aided treatments were developed along these dimensions. Furthermore, an assessment control group was formed.It was shown that performance improved significantly after treatment in comparison to the performance of the assessment control group. Comparison of the different training conditions showed that a number sentence treatment was the most effective training component. However, this effect was only present with the relatively high competent children and not with the relatively low performing children.     

中美学生数学问题解决的差异比较

“问题解决”是国际数学教育研究的一个热点,本文从中美两国学生在问题解决的过程和结果的比较研究入手,比较和研究了中美两国学生在问题解决中提出问题的能力及解决问题的策略两方面的差异。     

The consistency effect in word problem solving is effectively reduced through verbal instruction

In mathematical word problem solving, a relatively well-established finding is that more errors are made on word problems in which the relational keyword is inconsistent instead of consistent with the required arithmetic operation. This study aimed at reducing this consistency effect. Children solved a set of compare word problems before and after receiving a verbal instruction focusing on the consistency effect (or a control verbal instruction). Additionally, we explored potential transfer of the verbal instruction to word problems containing other relational keywords (e.g., larger/smaller than) than those in the verbal instruction (e.g., more/less than). Results showed a significant pretest-to posttest reduction of the consistency effect (but also an unexpected decrement on marked consistent problems) after the experimental verbal instruction but not after the control verbal instruction. No significant effects were found regarding transfer. It is concluded that our verbal instruction was useful for reducing the consistency effect, but future research should address how this benefit can be maintained without hampering performance on marked consistent problems.     

Teaching and testing mathematical problem solving by offering optional assistance

Transfer capability is usually defined as theability to apply acquired knowledge and skillsin novel situations.The experiment reported here concerned transferin mathematics education. An experimental programme was constructed, based on strengtheningthe connection of strategic and domain specificknowledge and offering hints during teaching aswell as during testing.Subjects were first graders from secondaryeducation in the Netherlands, from two schools,two classes each. Students from these fourclasses were randomly allocated either to theexperimental or the control group.The experimental computer-supported teachingprogramme was offered once a week during sixweeks; the control group received regularmathematics instruction.After controlling for the effects of thecovariates intelligence, mathematics aptitudeand anxiety, it was shown that the experimentalsubjects performed significantly better on aposttest than subjects in the control group.The results suggest that the experimentalinstruction method enhances mathematics problemsolving ability more strongly than traditionalinstruction. Upon closer examination thiseffect appears to be restricted to subjectsalready relatively high in intelligence andmathematical ability. This finding is notuncommon in intervention research, and issometimes referred to as the Matthew orthe fan-spread effect.     

有关数学问题的解题思想与方法

我们从中学就开始接触各类数学问题,而要解决这些数学问题,最重要的就是找出问题的精髓也就是所运用的思想与方法,并且这些思想与方法在实际应用中也非常广泛,因此,在这里我们主要介绍几种重要的解决数学问题的思想与方法。     

Middle school children's strategic behavior: Classification and relation to academic achievement and mathematical problem solving

Theories of problem solving (e.g., Verschaffelet al., 2000) hold strategic behavior centralto processing mathematical word problems. Thepresent study explores 80 sixth- andseventh-grade students' self-reported use of 14categories of strategies (Zimmerman &Martinez-Pons, 1986) and the relationship ofstrategy use to academic achievement,problem-solving behaviors, and problem-solvingsuccess. High and low achievement groupsdiffered in the number of different strategiesand categories of strategies reported but notin overall number of strategies, confidence inusing strategies, or frequency of strategy use.Students whose behaviors evidenced elaborationof the word problem's text reported moreself-evaluation; organizing and transforming;and goal setting and monitoring behavior.Implications for instructional practices thatsupport active stances toward problem solvingare discussed.     

中学数学中常用的解题思想和解题方法

阐述了中学数学中几种常用的解题思想：方程的思想、函数思想、转化思想、分类讨论思想、数形结合思想等.同时分析了几种常用的解题方法：换元法、消元法、参数法、递推法、构造法.     

问题解决式的缘起与理论背景

在改革高等数学教学的过程中,问题解决式教学是得到关注与认可的一种模式,然而,纵览相关的文献不难发现,介绍此教学模式的过程、方法的居多,而对这一模式兴起的原因以及理论基础进行分析的相对较少,这显然不利于此教学模式的推广.为此,本文从数学问题解决研究的兴起、教学模式的理论依据入手,介绍、分析了问题式高等数学教学模式背后的一些理论知识,期望以此推动新型数学教学模式的实施.     

Representation in memory and acquisition of knowledge from text and picture: Theoretical,methodological, and practical outcomes

Over the last 10 years, text comprehension research has undergone profound theoretical and methodological upheaval. Today, psychologists study language and texts because they present the trace markers of structures and cognitive operations. A text is no longer seen as a vehicle for conveying constructed meaning. Rather, text has been redefined as merely a structured sequence of stimuli, which activates both domain-related knowledge triggered by text information and linguistic knowledge. We present the theoretical views and experimental findings of the TEXTIMA group, characterized by the importance of the connectionist concepts.     

evolving research of mathematics teacher educators: the case of non-standard issues in solving standard problems

This study describes the learning of researchers who engage in mathematics teacher education as an integral part of their practice. As teacher educators working with teachers on the subject of proportional reasoning, the authors reflected on teachers solutions to a standard problem and analyzed answers that would conventionally be considered incorrect. This exploration showed that some incorrect answers made sense, were based on problem situation analysis, and brought attention to the fact that conventional formal answers were given without much deliberation on their meaning in the situation. This insight prompted a second research phase in which teachers discussed and explained alternative solutions, and developed deeper analysis of problem situation in solutions that had been correct in the first place. The importance of making teachers aware of the nature of alternative solutions was further exhibited in a third research phase in which teachers evaluated childrens answers to the same problem. The pedagogical insight that emerged stressed the importance of making teachers aware of the tension between an almost automatic application of a mathematical model, and of analyzing problem situation during problem solving. In addition, the researchers developed better understanding of the mathematical challenge associated with the proportional reasoning problem, a stronger awareness of the role of sensitivity to their learners (the teachers), and of the role of reflection.     

Impact of authenticity on sense making in word problem solving

The study presented in this paper seeks to investigate the impact of authenticity on the students’ disposition to make necessary real world considerations in their word problem solving. The aim is also to gather information about the extent to which different reasons for the students’ behaviors are responsible for not providing solutions that are consistent with the ‘real’ situations described in the word problems. The study includes both written solutions to word problems and interview data from 161 5th graders. The results show an impact of authenticity on both the presence of ‘real life’ considerations in the solution process and on the proportion of written solutions that were really affected by these considerations. The students’ frequent use of superficial solution strategies and their beliefs about mathematical word problem solving were found to be the main reasons for providing solutions that are inconsistent with the situations described in the word problems.     

小学数学学习困难学生加减应用题的补救教学

本研究采用小样本ABC实验设计 ,以 4名数学学习困难学生为被试评估了加减应用题补救教学的效果。本研究主要对数学学习困难学生进行了两个方面的训练 :(1)图式训练 ;(2 )一般解题策略训练。得到了以下结论 :(1)补救教学对学生加减应用题解题能力具有一定的促进作用 ;(2 )补救教学对学生的应用题解题行为产生了延时效果 ;(3)针对不同的学生 ,图式训练和解题策略训练存在不同的矫治效果。     

Teaching algorithmic problem solving or conceptual understanding: Role of developmental level,mental capacity,and cognitive style

It has been shown previously that many students solve chemistry problems using only algorithmic strategies and do not understand the chemical concepts on which the problems are based. It is plausible to suggest that if the information is presented in differing formats, the cognitive demand of a problem changes. The main objective of this study is to investigate the degree to which cognitive variables, such as developmental level, mental capacity, and disembedding ability explain student performance on problems which: (1) could be addressed by algorithms or (2) require conceptual understanding. All conceptual problems used in this study were based on a figurative format. The results obtained show that in all four problems requiring algorithmic strategies, developmental level of the students is the best predictor of success. This could be attributed to the fact that these are basically computational problems, requiring mathematical transformations. Although all three problems requiring conceptual understanding had an important aspect in common (the figurative format), in all three the best predictor of success is a different cognitive variable. It was concluded that: (1) the ability to solve computational problems (based on algorithms) is not the major factor in predicting success in solving problems that require conceptual understanding; (2) solving problems based on algorithmic strategies requires formal operational reasoning to a certain degree; and (3) student difficulty in solving problems that require conceptual understanding could be attributed to different cognitive variables.     

美国数学教育中的提出问题研究综述

在数学活动中,提出问题是指通过对情境的探索产生新问题,或在解决问题过程中对问题的再阐述.随着全美数学教师联合会<学校数学课程与评价标准>的颁布,提出问题成为美国数学课程的重要组成部分.最近20多年以来,美国在提出问题的教学研究方面取得了大量的研究成果.这些成果对于我国新课程下提出问题由课程理念向教学现实的转移具有借鉴和参考价值.     

Epistemic profiles and self-regulated learning: Examining relations in the context of mathematics problem solving

Relations were examined between epistemic profiles, regulation of cognition, and mathematics problem solving. Two hundred sixty-eight students were sampled from undergraduate mathematics and statistics courses. Students completed inventories reflecting their epistemic profiles and learning strategies, and were profiled as rational, empirical, or both. Based on their profiles, 24 students participated in two problem-solving sessions. Episodes were coded for planning, monitoring, control, use of empirical and rational argumentation, and justification for solutions. For both self-reported metacognitive self-regulation and regulation of cognition during problem solving, students profiled as rational had the highest self-reported mean and actual frequency of regulation of cognition compared to students profiled as predominantly empirical. Moreover, students profiled as predominantly rational correctly solved more problems than the other two groups. Finally, students’ approaches to problem solving were consistent with their epistemic profiles. Relations are discussed in the context of various theoretical frameworks.     

The mathematical discourse of 13-year-old partnered problem solving and its relation to the mathematics that emerges

This paper, written within a discursive perspective, explores the co-shaping of public and private discourse, and some of the circumstances under which one occasions the other, in the evolution of mathematical thinking by pairs of 13-year-olds. The discourse of six pairs of students, engaged in interpreting and graphing problem situations involving rational functions, was analyzed by means of recently developed methodological tools. The nature of the mathematics that emerged for each pair was found to be related to several factors that included the characteristics of the interpersonal object-level utterances both before and after the solution path had been generated, the degree of activity of the personal channels of the interlocutors, and the extent to which the thoughts of participants were made explicit in the public discourse. The analysis of the discursive interactions provided evidence that adolescents within novel problem situations can experience some difficulty in making their emergent thinking available to their partners in such a way that the interaction be highly mathematically productive for both of them. This revised version was published online in July 2006 with corrections to the Cover Date.     

职业教育集团化办学的理论分析与实践探索

职业教育集团化办学是当前职业教育改革发展的新举措。为科学有效地推行职业教育集团化办学,首先要明白职业教育集团化办学的理论基础和现实不足。并在此基础上,从多层面、多角度去探索职业教育集团化办学的实施路径。本文基于职业教育集团化办学的理论分析与现状分析,从外部环境的营造、合作对象的选择、各相关主体的利益平衡、组织建设和制度建设等方面,进行了职业教育集团化办学的路径分析与探索。     

基于问题解决的中国数学教育研究七十年:价值嬗变与研究展望

在过去的70年里,问题解决一直是我国数学教育领域的研究热点,其成果不仅影响着学生高层次思维的发展,还促进了积极的学习态度。基于问题解决的数学教育研究历程可分为三个阶段:初兴阶段、发展阶段和深化阶段。问题解决在不同阶段的名称反映了不同时期的价值追求。认知结构研究的抽象化、过程模型研究的多元化、策略研究的高度概括以及元认知研究的外显是数学问题解决研究的趋势。展望未来,关注同一情境中的不同结构、同一结构在不同情境间的迁移,为知识、技能向问题解决能力的转化匹配学习条件,加强数学问题解决的表现性评价研究是今后的研究方向。     

中小学数学情境与提出问题教学探析

中小学“ 情境—问题” 数学教学研究是基于国际、国内数学教育改革大趋势而开展的一项前沿性、探索性与实证性的数学教学研究.它所提倡的课堂教学观有:(1)以人为本,以学生为中心;(2)关注学生的学习过程;(3)发展共性与突出个性.“ 情境—问题” 数学教学实验研究还应在以下 3 方面做出努力:(1)实验研究范围需进一步扩大;(2)将“ 情境—问题” 数学教学的基本理念向其它学科渗透;(3)进一步提升理论研究水平.     

中小学"数学情境与提出问题"教学的实验研究

This research tends to make the experimental study on the mathematics teaching model of “situated creation and problem-based instruction” (SCPBI), namely, the teaching process of “creating situations—posing problems—solving problems—applying mathematics”. It is aimed at changing the situation where students generally lack problem-based learning experience and problem awareness. Result shows that this teaching model plays a vital role in arousing students’ interest in mathematics, improving their ability to pose problems and upgrading their mathematics learning ability as well.